is either zero or full if the sum \(\sum _{q=1}^\infty q\Psi (q)\)  converges or diverges, respectively, see [1]} for a statement of the modern version of this theorem. The monotonicity of \(\Psi \)  is hereby only needed
for the divergence part. In fact, Duffin-Schaeffer (1941) constructed a counter example to show that without the monotonicity assumption Khintchine's theorem is false. However, they conjectured a refined statement that holds
for non-monotonic \(\Psi \)  as well, which has been recently affirmatively proved by Koukoulopoulos-Maynard [2]}.
